Biomedicine: A fertile, challenging and worthy field for ......Biomedicine: A fertile, challenging...

Biomedicine: A fertile, challenging and worthyfield for mathematical and engineering research

K. R. Rajagopal

Texas A&M UniversityCollege Station, Texas 77845

”Most of what is characteristic of living organisms cannot beexpressed in mathematical terms of the simplistic laws of physics.”

– E. Mayr

”The existence of life must be considered as an elementary factthat cannot be explained, but must be taken as a starting point inbiology, in a similar way as the quantum of action, which appearsas an irrational element from the point of view of classicalmechanical physics, taken together with the existence ofelementary particles, forms the foundation of atomic physics.”

– N. Bohr

”Surely no biologists would even express such a hope. It would bedifficult to expect the incredible diversity of nature, the complexityof the process of ontogenetic differentiation and of the nervoussystem, or the qualitative uniqueness of each kind ofmacromolecule, could be expressed in the form of a few simplegeneral laws.”

– E. Mayr

K. R. Rajagopal Biomedicine: A fertile, challenging and worthy field for mathematical and engineering 2/42

– E. Mayr

– N. Bohr

– E. Mayr

– N. Bohr

– E. Mayr

I will confine my discussion to some issues in

1 Cardiac Mechanics

And if time permits to some issues concerning

2 Growth and Remodeling

1 Cardiac Mechanics

Pathologies of Clot formationArterial MechanicsAortic Dissection

Cardiac Mechanics

Pathologies of Clot formation and Lysis:Causes for thrombosis (Virchow Triad)

1 Local flow stasis/stagnation

2 Blood Vessel Injury or Endothelial Dysfunction3 Hypercoagulability (an augmented native tendency of blood to

form clots)

Disorders of pathologic thrombus formation and maintenanceDisorders characterized by impaired thrombus formation andMaintenance

Cardiac Mechanics

form clots)

Cardiac Mechanics

form clots)

Cardiac Mechanics

form clots)

Cardiac Mechanics

form clots)

1 Atrial thrombosis

Caused mainly due to atrial dysrhythmias (atrial fibrillationand atrial flutter)There is local flow stagnation leading to atrial thrombusformation.

2 Ventricular thrombosis

Mainly due to severe systolic ventricular dysfunction andventricular aneurysms characterized by regional ventricular walldilation and thinning that is associated with paradoxicalexpansion during ventricular systole that is associated with ahigh rate of intra-cavitary thrombus formation.

3 Mostly a problem with artificial mechanical valves due tonon-endothelialized surfaces.

1 Atrial thrombosis

4 Arterial thrombosisArterial insufficiency or impaired local arterial blood flow(ischemia) and oxygen delivery.

Acute coronary syndromes: Thrombus formation over unstableplaque, critical stenosis reached by a stable plaque, coronaryvasospasm and acute increase in myocardial oxygenconsumption demand.Extremity arterial insufficiency: Due to thrombus formationover unstable plaque in patients with pre-existentatherosclerotic disease. Formation of thrombo- or atheroembolism.

5 Capillary thrombosis

Not well understood. Associated with disseminatedintravascular coagulation.

6 Venous thrombosis and Pulmonary thrombo-embolismGenetic disorders in which coagulation factors are synthesizedin excessive amounts or anti-coagulant or fibrinolytic factorsare synthesized in inadequate amounts.

Factor V LeidenMutant prothrombinProtein C deficiencyProtein S deficiencyAT-III deficiencyEndothelial dysfunction or injury

Bleeding Disorders

(a) Platelet disorders: Thrombocytopenia (decreased bloodplatelet concentration) Platelet count ≤ 20000. Manyiatrogenic processes lead to platelet disorders. Also,endogenous disease states lead to platelet dysfunction.

(b) Disorders of Coagulation factors and Fibrinolysis: Reducedlevel of activity of coagulation factors. Pathologically activatedfibrinolysis. Factor deficiencies.

Hemophilia AHemophilia BLiver failure (Depression of coagulation factor levels)Disseminated intravascular coagulation

Bleeding Disorders

Composition of Whole Blood

Cell matter ≈ 46 % by volume of blood

(RBC) erythrocytes: ≈ 98 % of cell matter

(WBC) leukocytes

Platelets

Plasma is primarily water (92 - 93 %) in which variousproteins are dissolved along with various ions.

Proteins: f-I: brinogen, f-II:prothrombin, f-V, f-VIII, f-IX, f-X,f-XI, f-XII, f-XIII, antithrombin III, Tissue factor pathwayinhibitor, protein C, protein S, Plasminogen, α1-antitrypsin,α2-antiplasmin

Ions: Na+,K+, Ca2+,Mg2+, Cl−, HCO−3 , PO3−4

Thus blood is a very complex mixture.

(WBC) leukocytes

Platelets

(WBC) leukocytes

Platelets

(WBC) leukocytes

Platelets

(WBC) leukocytes

Platelets

(WBC) leukocytes

Platelets

(WBC) leukocytes

Platelets

(WBC) leukocytes

Platelets

(WBC) leukocytes

Platelets

Q: Is it reasonable to treat blood as a single component fluid?

A: Depends on the application.

Q: Is it reasonable to treat blood as a single component fluid?

A: Depends on the application.

Plasma - Newtonian fluid ≈ 1.2 cP.

Erythrocytes (RBCs): biconcave deformable discs that lacknuclei.

Membrane: approximately 3 % by weight of RBC

cytoplasm - solution of hemoglobin in water - viscoelastic(Evans and Hochmuth, 1976; based on micropipetteexperiments).

Leukocytes : approximately 1 % volume of blood.

Granulocytes - viscoelastic (Schmidschonbein and Sung,1981).

Monocytes

Lymphocytes

Platelets: Elastic/Viscoelastic

Monocytes

Lymphocytes

Monocytes

Lymphocytes

Monocytes

Lymphocytes

Monocytes

Lymphocytes

Monocytes

Lymphocytes

Monocytes

Lymphocytes

Monocytes

Lymphocytes

Monocytes

Lymphocytes

Models from the perspective of hemodynamics

Homogenized Fluid Models

Newtonian Fluid: May be reasonable in large vesselsGeneralized Newtonian Fluid: Necessary in smaller vesselsViscoelastic fluid capable of shear thinning and relaxation timedepending on the shear rate.Reviews of One dimensional continuum models - Cho andKensey (1991)Three dimensional continuum models - Yeleswarapu (1996)

Models based on mixture theory

Mixture models - Trowbridge (1984); Kline (1972); Chathuraniand Upadhya (1979)

We also need to take into account:

The complex biochemical reactions that take place.

Shear-thinning property of blood Charm and Kurland, 1965;Chien et al., 1966;

Stress-relaxation Thurston, 1972

Viscoelasticity of RBC membrance Evans and Hochmuth,1976

Variation of stress-relaxation with shear rate Thurston, 1973

Numerous generalized Newtonian uid models have beenproposed for blood.

Rate-type fluid model for blood: Thurston, 1972; Yeleswarapuet al.,1998

Platelet Activation and Aggregation

Platelets constitute approximately 3 % of blood - discoid cellfragment - approximately 6 µm3.

Platelet activation occurs due to interaction with collagensand adhesive glycoproteins exposed by damage with thrombinor adenosine diphosphate (ADP).

Activation - Organelles within the platelet are centralized. -Glycoproteins on the platelet membrane undergo a change inconformation.

Pseudopods are extended so that the platelet is a sticky spinysphere.

Platelet activation is followed by interaction with plasmaproteins like Factor IX, Factor V, vWF, fibrinogen and fibrinso as to adhere to sub-endothelial tissue and leads toaggregation.

Intrinsic Pathway to Coagulation: Factor XII→Fibrin

Extrinsic Pathway to Coagulation: Factor VII→Fibrin

Starts with the exposure of TF (a cell membrane boundprotein) in the subendothelium to blood which leads to achain of coagulation reactions.

Formation of TF-VIIa→Enzymes, Factor IXa, Xa→Factor Vaand VIIIa

The enzyme complex IXa-VIIIa bound to the membrane of theactivated platelet catalyzes the formation of Xa from X.

Next, the enzyme complex Xa-Va formed on the membrane ofthe activated platelet catalyzes the production of thrombinfrom prothrombin. Thrombin acts on fibrinogen to yield fibrinmonomers that polymerize and are cross-linked to form afibrin matrix.

Three inhibitory mechanisms in blood:

AntiThrombin III (AT III), TF Pathway Inhibitor (TFPI), APCBauer and Rosenberg, 1995, November, 1995.

Fibrinolysis and Clot Dissolution

Enzymatic reactions initiated when thrombin and fibrinactivate endothelial cells resulting in enhanced production oftissue PLS activator (tPA) and urokinase-like PLS activator(uPA).

tPA and uPA catalyze the transformation of PLS into activeenzyme plasmin.

Plasmin degrades the fibrin polymer into smaller units leadingto dissolution of the clot.

Fibrinolysis has its share of regulatory mechanisms. We shallnot discuss these in detail.

Clot dissolution can also occur due to high shear stress whichcauses the fibrin to rupture. (See Riha et al., 1999).

Model Characteristics

Whole blood is modeled as a shear thinning viscoelastic fluidthat contains the various reactants involved in clot formation.

Development of convection-reaction-diffusion equations thatgovern the generation/depletion of the reactants.

Platelet activation either due to action by thrombin andagonists like ADP or prolonged exposure to shear stresses.

Boundary conditions that represent the level of injury andendothelial cell activity

Prescription of a threshold concentration of surface boundTF-VIIa Complex.

A model for the clot as a viscoelastic fluid with a muchgreater viscosity than pure blood.

Clot growth, the boundary defined by a fibrin concentration.

Clot dissolution due to decrease in fibrin concentration andshear stresses.

Basic Kinematics

κτ ,κt- Placersκτ (B),κt(B) - ConfigurationsMotion is a one-parameter familyof placers.X ∈ B, X = κR(X) ,x = κt(X)

Identify motion by

x = χκR(X, t).

ξ = χκR(X, τ) = χκR(χ−1κR

(X, t), τ)

= χκt(x, τ)

...Relative Motion

Deformation Gradient

F κR :=∂χκR∂X

F κR is a linear transformation from the tangent space at X to thetangent space at xRelative Deformation Gradient

F κR :=∂χt∂X

φ = φ(X, t) = φ(x, t) (3)

5φ :=∂φ

∂X, gradφ :=

∂x, (4)

∂t:=

∂t(5)

Lagrangean Eulerian

CONFIGURATION OF A BODY

Figure: κp(τ) - Natural configuration corresponding to κτκp(t) - Natural configuration corresponding to κt

If one inhomogeneously deforms a body and then removes thetraction, it is possible that the unloaded body will not fittogether compatably and be simultaneously stress free in anEuclidean space.

However, it can be unloaded in a non-Euclidean space inwhich it fits together and is stress free (Eckart 1940S)

However, a ”sufficiently small” neighborhood of a materialpoint can be unloaded to a stress free state in Euclideanspace, i.e., if the deformation is reasonably smooth, we canpick sufficiently small neighborhoods wherein the deformationis homogeneous. The notion of a configuration really appliesto an appropriately small neighborhood of a point.

Henceforth, for the sake of illustration, let us assumehomogeneous deformations.

Natural Configuration– Can think of it as a stress-free configuration– It is a ”local notion”.– It is really an equivalence class of configurations.

Eg: Classical Plasticity

Figure: Traditional Plasticity

Classical metal plasticity involves an infinity of naturalconfigurations, and to determine the stress we requirekinematical information from more than one naturalconfiguration.

The response is elastic from each of these naturalconfigurations and the inelasticity is purely due to the changein the natural configurations.

Classical metal plasticity involves an infinity of naturalconfigurations, and to determine the stress we requirekinematical information from more than one naturalconfiguration.

The response is elastic from each of these naturalconfigurations and the inelasticity is purely due to the changein the natural configurations.

Eg. 2 TwinningIn twinning there are a finite number. As many as the numberof variants.

Figure: Modulo variants, we have two natural configurations, thatcorresponding to O and F, and these two natural configurations havedifferent material symmetries.

Other Examples:

Viscoelasticity

Superplasticity

Crystallization

Multi-network Polymers

Classical theories are trivial examples:In classical elasticity the natural configuration does not evolve.In classical fluids the current configuration is the naturalconfiguration.

Figure: Configuration as a localnotion

Figure: Spider spinning a web

New material is laid in a stressed state. It can have a differentnatural configuration than the material laid down previously.

Restrict ourselves to homogeneous deformation.Think in terms of Global configurations.

Figure: Non-uniqueness of stress-free state (Modulo rigid motion)

More than one Natural Configuration can be associated withthe current deformed configuration.Example: Consider a Viscoelastic body capable ofinstantaneous elastic response κt

– Natural Configuration reached by instantaneous unloading–Anadiabatic process.

– Natural Configuration reached in an isothermalstress-relaxation process

Thus, we need to know the process class under consideration.

SINGLE CONSTITUENT BODY

Balance of Mass:∂ρ

∂t+ div(ρv) = 0 (6)

Assumption of the incompressibility implies that the body canundergo only isochoric motion, i.e.,

div(v) = 0 (7)

Balance of Linear Momentum:

divT+ ρb = ρdv

SINGLE CONSTITUENT BODY

Balance of Angular Momentum:

T = TT (9)

Balance of Energy:

dt+ divq− T.L− ρr = 0 (10)

2nd Law:

dt+ div

θ− ρr

θ:= ρξ >= 0 (11)

T-stress, η - specific entropy, θ-temperature,q-heat flux vector,r-radiant heating

The evolution of the natural configuration, amongst otherthings, is determined by the maximization of the entropyproduction.

Ziegler suggested the use of maximization of dissipation, butnot within this context.

The maximization of the entropy production makes choicesamongst possible reponse functions. For instance, it will picka rate of dissipation(or entropy production) from amongst aclass of candidates.

For a class of materials, such a choice leads to a Liapunovfunction that decreases with time to a minimum value(Onsager/Prigogine- Minimum entropy production criterion).Rajagopal and Srinivasa(2003), Proc. Royal Society.

There is no contradiction between these two criteria:– Maximization of the entropy production to pick constitutive

equations and minimization of entropy production with timeonce a choice has been made. (Rajagopal andSrinivasa(2002)).

During the process entropy is produced in a variety of ways:1 Due to conduction2 Due to mixing3 Due to work being converted to heat (dissipation)4 Phase change5 Growth

Part of the energy that is supplied to the body is stored in thebody in a variety of ways. The energy supplied

1 Can change the kinetic energy.2 Can change the potential energy.3 Is stored as ”strain energy”

(a) that can be recovered in a purely mechanical process(b) that can only be recovered in a thermal process.

– Part of the energy due to mechanical working is transferred asenergy in its thermal form (Heat).

– Part of the energy changes the ”Latent Energy”.– Part goes towards ”Latent Heat”

Figure: Biochemical model of selected reactions involved in the extrinsicand intrinsic coagulation pathways and fibrinolysis. Arrow heads with aplus sign near them indicate activation or enzymatic stimulation. Arrowheads with a minus sign near them indicate inactivation or inhibition.

Table: Scheme of enzymatic reactions.

The constitutive equations derived using the constrainedmaximization procedure are

T = −p1+ S,

S = µBκp(t) + η1D,

5Bκp(t) = −2µ

(Bκp(t) − λ1

trB−1κp(t).

The biochemical reactions are given by:

∂[Yi]

∂t+ div {[Yi]v} = div {DYi(D)[Yi]}+GYi

Here, and elsewhere below, [Yi] represents the concentration of thereactant Yi; GYi represents the production or depletion of Yi dueto the enzymatic reactions, v is the velocity, and DYi representsthe diffusion coefficient of Yi which could be a function of theshear rate (captured by means of the stretching tensor D).It is also possible to allow for GYi to depend on the concentrationof the jth reactant.

Model for ClotSimilar form as that for blood. However, the material functions aredifferent. They reflect the fact that the clot is much more viscousthan whole blood.

Activation Criterion:

A(t) = A(0) +1

k(|Trz |Tth

)H (|Trz| − Tth) dt.

The Criterion for the Activation of Resting Platelets [RP]:If A (t− tact) > Athr and A(t) < Adam orA (t− tact) = Athr, A (t− tact) > 0 and A(t) < Adam, then

[AP ](t) = [AP ] (t− tact) + [RP ] (t− tact) .

The Criterion for Lysis of PlateletsIf A (t− tact) > Athr and A(t) > Adam then

[AP ](t) = [RP ] (t− tact) .

Model for Dissolved Clots:We assume the clot on dissolution reverts to the original model forwhole blood.

Figure: Pressure gradient components in-phase with (P’), andout-of-phase with (P”) the rms volumetric flow rate (U) duringoscillatory flow in small tubes.

Figure: Velocity profiles: Theoretical predictions, for Poiseuille flow, usingthe proposed model, the GOB model, and the GM model, are comparedwith the data for porcine blood.

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